GLOBAL UNRAMIFIED GEOMETRIC LANGLANDS CONJECTURE

WITTEN-KAPUSTIN

MANU CHAO

Contents

1. Global Unramified Geometric Langlands conjecture 2

1.1. Geometric Langlands conjectures 2

1.2. The categorical meaning of Hecke operators 4

1.3. Geometric Langlands conjectures for line bundles 5

1.4. Hecke modifications and the loop Grassmannian 11

1.5. Hecke operators and the stratifications of the loop Grassmannians 14

1.6. Examples in type A 21

1.7. Langlands conjectures in number theory 21

1.8. Appendix. Dual group G comes with extra features 25

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2 MANU CHAO

1. Global Unramified Geometric Langlands conjecture

1.1. Geometric Langlands conjectures. These are algebro-geometric expectationswith roots in Number Theory. Here global means that conjectures concern a projec-tive curve over a closed field while unramified means that we consider local systems whichare defined on the whole curve, i.e., have no singularities. The conjectures have twomain forms – (a) categorical equivalence, and (b) existence of Hecke eigensheaves.

1.1.1. The data. Let C be a smooth projective curve over a closed field k. Choose areductive algebraic k-group G, it turns out that such groups come in Langlands dual pairsG, G. We will attach to each of these groups one moduli of objects on C and a categoryof sheaves on this moduli. We will be interested in the case k = C which allows thedifferential geometric treatment of Kapustin and Witten. For the general k the story istechnically different as it uses the language of l-adic cohomology.

First, let BunG(C) be the moduli stack of G-torsors (principal G-bundles) over C. IfG = GLn then G-torsors are equivalent to rank n vector bundles, so BunG(C) is the sameas the moduli Vecn(C) of such vector bundles on C.

Next, the moduli stack LSG(C) of G-local systems on C classifies (if the characteristic ofk is zero), pairs of a G-torsor P and a connection ∇ on P (by that we mean a G-invariantconnection).(1) If k = C we can think of G-local systems as group maps π1(C)→ G. Wewill attach this moduli to G, so we get LSG(C).

1.1.2. Conjectures. The Geometric Langlands conjecture of Beilinson and Drinfeld saysthat G-local systems on C can be encoded as D-modules on the stack moduli BunG(C).We denote by DX the sheaf of algebraic differential operators on X.

In the deeper categorical form, the moduli LSG(C) and BunG(C) are related by a kind ofa Fourier transform exchanging coherent sheaves on LSG(C) and D-modules on BunG.The fact that the duality also exchanges G and its Langlands dual G is alike the fact thatthe Fourier transform exchanges functions on dual vector spaces.

1.1.3. Conjecture. (a) There is a canonical equivalence of triangulated categories

L : Db[Coh(LSG(C))] ∼= Db[mod(DBunG(C))].

We call L the Langlands transform.

(b) For any G-local system E on the curve C, there is a unique Hecke eigen-D-module Eon BunG(C), with eigenvalue E .

1For arbitrary k there is still a notion of a G-local system over C, these are the l-adic local systemsin etale topology. In this case we use the Ql-form GQl

of G, so for G = GLn a G-local system meansa ordinary rank n l-adic local system. For a general G a G-local system L is then viewed in Tannakianterms as a family of ordinary l-adic local systems LV indexed by representations V of G

Ql, i.e., LV

plays the role of associated bundle for a GQl-torsor L.

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(c) In the setting when both conjectures make sense, they are related by: E is theLanglands transform L(OE) of the structure sheaf of the point E in LSG(C).

1.1.4. Remarks. (0) Some of this is imprecise, for instance a question of the correct cate-gory on LSG(C).

(1) Part (b) is the original Geometric Langlands conjecture of Drinfeld which is a directgeometrization of a Langlands conjecture in Number Theory. It makes sense over anyclosed field k since it involves only the stack BunG(C), however the original interest wasin the case when k is the algebraic closure over a finite field since this is relevant forNumber Theory. For G = GLn and k = C this conjecture has been proved by Gaitsgory.

(2) Claim (a) is a lift of (b) to a categorical statement As we will see, if we know or believe(a) then the claim (b) roughly says that the Langlands transform exchanges tensoring ofcoherent sheaves on LSG(C) and convolution of D-modules on BunG(C). (Recall that theusual Fourier transform exchanges multiplication and convolution of functions.)

However, (a) contains much more information since it deals with a finer structure of acategory. It is also “more reasonable” in the sense that it involves fewer details. (a) hasnot been predicted from Number Theory but should contain information of interest forNumber Theory. (For instance the isomorphism of K-groups of categories?)

(3) The more general ramified version deals with local systems on a punctured curve, i.e.,local systems on C with singularities at finitely many points. For this generalization onemodifies BunG(C) by adding parabolic structures at singular points. Not much is knownin mathematical literature, though there are some results for P1. Recent contributionfrom physics (Gukov-Witten) seems significant.

1.1.5. Dependence on the field k. In the case k = C that we will be interested in all thesespaces are complex manifolds (stacks) and physicists find that they are a part of a largerdifferential geometric picture which lives in real manifolds.

For general k, while the stack BunG(C) is defined over k, the moduli LSG(C) of l-adic localsystems is a stack over Ql. Also, D-modules on BunG(C) have to be replaced by perversel-adic sheaves on BunG(C). So, the conjectured equivalence concerns two triangulatedcategories which are Ql-linear.

1.1.6. Of course, we need to explain the meaning of Hecke eigen-D-modules which wecall simply Hecke eigensheaves. Contrary to history, we will start by “guessing” (b) fromthe formal structure of (a), and then we will make these ideas precise by going through theabelian case G = Gm which is a reformulation of standard facts about Jacobian varieties(this is of course the historical development from abelian to nonabelian).

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1.2. The categorical meaning of Hecke operators. Hecke eigensheaf property turnsout to be quite reasonable – just the Fourier transform of the most obvious description ofpoints in space, i.e., of the characterization of structure sheaves of points.

1.2.1. Eigenvalue property of points. Conjecture (a), says that we have two equivalentsettings BunG(C) and LSG(C). A G-local system E is a point of LSG(C) and since weknow something about Db[Coh(LSG(C)] the question arises how does one describe a pointa of a scheme X in terms of the category Db[Coh(X)]? The simplest answer is that thestructure sheaf of a point behaves simple under tensoring with a vector bundle V – weget a multiple of Oa with multiplicity given by the fiber of V at a :

V ⊗OXOa∼= Va ⊗k Oa.

We can say that Oa is a common eigensheaf for operators V⊗OX−, where V goes through

all vector bundles on X. Notice that our “operators” are really functors and the corre-sponding “eigenvalue” is a functor from vector bundles to vector spaces – the fiber of ata.

1.2.2. Tautological vector bundles on LSG(C). To see what the above philosophy meansfor a point E in LSG(C) we need to know some vector bundles on LSG(C). OverLSG(C)×C there is a tautological G-torsor P, for any local system E = (P,∇) therestriction of P to E×C is the underlying G-torsor P . Of course, P is really the tautolog-ical family of local systems over LSG(C) and it carries a connection in the direction of C,i.e., a relative connection for the projection to LSG(C).

Now, any representation V of G defines a vector bundle Vdef= P×GV on LSG(C)×C. In

particular, any point a ∈ C defines a vector bundle Va on LSG(C) – the restriction of V

to LSG(C)×a. Therefore, OE is an eigenvector for the tensoring operatorsWa,Vdef= Va⊗−

:

Va⊗OLSG

(C)OE∼= (Va)E⊗k OE ,

and the eigenvalue for the tensoring operator Wa,V is the fiber (a vector space)

(Va)E = Pa×G V = Ea×G V,

where Ea = Pa is the fiber of the local system E at a ∈ C.

Remarks. (1) Instead of using one point of C at a time, one could also write a singleequation over LSG(C)×C

V ⊗OLSG

(C)×COE×C

∼= V |E×C ⊗OCOE×C

for each V inRep(G). Here C is the parameter space for operators and the eigenvalue istherefore a vector bundle on C (actually a local system!)

(V )E×C = E×G V.

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(2) Here we use of all representations V of G, i.e., all conversions of the G-local systemE into a local system of vector spaces, in order to be able to tensor. So, the meaningshould really be (in a correct categorical framework)

P ×G OE×C∼= E ×G OE×C ,

so the eigenvalue is really the local system E itself.

1.2.3. Algebra of tensoring operators. Notice that at each point a ∈ C our operators forma copy of the tensor category (Rep(G),⊗k) since clearly

(1) Wa,U⊕V =Wa,U⊕Wa,V and(2) Wa,U⊗V =Wa,UWa,V .

Observation (1) implies that it is sufficient to describe these operators for irreducible V ’s.

1.2.4. Langlands transform as a Fourier transform. Since we are dealing here with “dual”groups, we may hope that the transform L is somewhat like the Fourier transform. Theusual F-transform exchanges multiplication and convolution of functions. So, one mayhope that the Langlands transform will exchange tensoring which is alike multiplication (itis done pointwise!), with something like convolution. On the coherent side the structuresheaf OE is an eigenvector of tensoring operators Va⊗O− indexed by all V ∈ Rep(G)and points a ∈ C, and the eigenvalue is E . Then on the D-module side LE should becharacterized as a common eigenvector with eigenvalue E for a family of operators thatare (roughly) convolutions with Langlands transforms of tautological bundles. These newoperators will be called Hecke operators Ha(V ) (once they are defined), they will also beparameterized by the same data V, a.

Next, we check that all of this really works when G = Gm and we use this case to guesswhat Hecke operators should be for general G.

1.3. Geometric Langlands conjectures for line bundles. Here, we notice that forG = Gm Langlands conjectures amount to classical facts about the Jacobian, We start bynoticing that in this case the relevant moduli are classical objects. Then the categoricalequivalence is proclaimed to be a version of the Fourier-Mukai transform based on theself-duality of the Jacobian.

Now, we come to our real subject – the meaning of Hecke operators for G = Gm. Asthe passage from a curve to its Jacobian is a form of linearization where a geometricobject is encoded in an abelian group, the same is true for a passage from a local systemE on C to the corresponding D-module E on the Jacobian (actually just a local systemon the Jacobian). So, the characterizing property for E is its compatibility with thegroup structure on the Jacobian. It turns out that we can reduce this property to itsminimal formulation as compatibility of E with the modification operators on line bundlesHa : L7→L(a) defined at each point a ∈ C. The upshot is that the characterizing propertycan be stated as an eigenvector property for the operators Ta, a ∈ C, which act on local

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systems on BunG(C) by the pull-backs with respect to the modification operators Ha.

These are the Hecke operators Tadef= H∗

a .

1.3.1. Moduli of line bundles. A torsor P for G = Gm is the same as a line bundle L.Here P is obtained from L by removing the zero section and L = P×Gm

k. The standardversion of the moduli of line bundles is the Jacobian JC which is an algebraic ind-group (itsconnected components are abelian varieties). However, each Gm-torsor (or line bundle)on C has automorphism group Gm and therefore the stack moduli BunGm

(C) is slightlymore subtle:

BunGm(C) ∼= JC/Gm = JC×

pt

Gm,

for the trivial action of Gm on JC .

Let me mention some of the familiar aspects of Jacobians:

(1) The connected components JnC of the Jacobian are indexed by the degree n ∈ Z

of the line bundle.(2) (Abel-Jacobi maps.) For n ≥ 0 there is a canonical map Cn → Jn

C by(a1, ..., an)7→ O(a1 + · · · + an). It factors through the nth symmetric power of

the curve C(n) def= (Cn)//Sn (the categorical quotient), which is the moduli of

effective divisors of degree n. This yields the Abel-Jacobi maps AJn :

Cn qn−→ C(n) AJn

−−→ JnC , AJn(D) = O(D).

(3) (Abelian group.) Jacobian is the commutative group ind-scheme freely generatedby the algebraic variety C. What this means is that there is a canonical mapAJ1 : C → J1

C and it is universal among all maps of C into commutative ind-algebraic groups, i.e., any such map C → A factors uniquely through JC .(2)

(4) (Self-duality.)(3) J0C is an abelian variety which is canonically self-dual.

1.3.2. Moduli of invertible local systems. Local systems for G = Gm are the same asinvertible local systems, so we denote the moduli by ILS(C) ∼= LSGm

(C). There is aforgetful map

ILS(C)F−→ JC ,

since an invertible local system is a pair E = (L,∇) of a line bundle L and a connection ∇

on L, The map is really ILS(C)F−→ J0

C because a line bundle that carries a connectionnecessary has degree 0, The fiber of this forgetful map at the trivial line bundle , i.e., allconnections on the trivial line bundle is natural identified with the vector space Γ(C, ωC)of global 1-forms. More generally, the connections on any line bundle L ∈ J 0

C are a torsor

2The correspondence C 7→JC is the origin of Grothendieck’s idea of motives – one should associate toalgebraic varieties commutative ind-groups which contain most of the information about the variety byTorelli theorems.

3This is one many formulations of Class Field Theory.

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for Γ(C, ΩC) since any two connections differ by a 1-form. So ILS(C) is a torsor for thevector bundle J0

C×Γ(C, ωC) over J0C .

The (co)tangent bundles over JC are trivial vector bundles since JC is a group, SoTJC

∼= JC×H1(C,O) because the tangent space at 1 is the linearization H1(C,O) ofJC = H1(C, Gm). By Serre duality T ∗JC = JC×Γ(C, ΩC). Therefore, ILS(C)→ J0

C is atorsor for T ∗J0

C .

Notice that adding a connection ∇ to a line bundle L reduces the the automorphismgroup Gm of L.

1.3.3. Categorical Geometric Langlands conjecture for Gm. The derived category of co-herent sheaves on BunG(C) = JC/Gm is the derived category of Gm-equivariant, i.e.,graded, coherent sheaves on the Jacobian :

Db[Coh(BunGm(C))] = Db[CohGm

(JC)].

On the other hand, we have D-modules on LSGm(C) = ILS(C), which is a torsor for

T ∗J0C .

So, the categorical part of the Langlands conjecture for Gm says that

Theorem. There is canonical equivalence

Db[CohGm(JC)] ∼= Db[mod(DILS(C))].

“ Proof. ” The underlying geometric content here is that the abelian variety JC iscanonically self-dual. Then the equivalence above can be see to be a “twisted” versionof the Fourier-Mukai equivalences. In local systems the twist comes from replacing thegroup T ∗J0

C by its torsor ILS. The corresponding twist on the side of G-bundles is givenby replacing coherent sheaves with D-modules. This twisted version is then just a matterof standard group theory.

Historically, Fourier-Mukai equivalence for dual abelian varieties has been a prototypefor Laumon’s Fourier transform of 1-motives, and the equivalence above is essentially aconsequence of Laumon’s point of view. The best treatment is in [Polishchuk-Rothstein].

1.3.4. Correspondence of invertible local systems on the curve and on its Jacobian. ForG = Gm, the geometric Langlands correspondence should assign to each invertible localsystem E on C a D-module LE on BunGm

(C) = JC×pt/Gm which is an “eigensheaf forHecke operators”. It will turn out that in this case D-module LE is again an invertiblelocal system on JC and the whole picture – a correspondence of invertible local systemson C and on JC – is embedded in the standard theory of Jacobians.

To find the mechanism of the correspondence we start with the case k = C where we canuse the complex (i.e., transcendental) methods. Then we will see a geometric proofs validfor general k.

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1.3.5. Characterization of the correspondence E 7→E . Over C it is well known that π1(JnC)

is H1(C, Z), i.e., the maximal abelian quotient of π1(C, Z). Invertible local systems arejust the one dimensional representations of fundamental groups, so for each n ∈ Z thereis a bijection of invertible local systems E on C and invertible local systems on Jn

C . Forn = 0 we denote it E 7→L0

E .

The remaining discrepancy is that there are more invertible local systems on JC thenon C simply because of the disconnectedness of JC . From this point of view, a naturalcorrespondence E 7→LE would follow from any canonical extension procedure L0 7→L ofinvertible local systems L0 from J0

C to JC .

Question. What should be a characterizing property of such procedure?

The basic property of the Jacobian is that (i) it is an ind-algebraic group, (ii) which isfreely generated by C. So,

• From the point of view of (i), the characterizing property of the extension shouldbe the compatibility with the group structure of JC .• (ii) Moreover, from the point of view of (ii) we can play with the formulation of

this property by sometimes replacing JC with just C.

The first approach is simpler but is is the second one that will generalize into the notionof Hecke operators for all groups G.

1.3.6. Compatibility with the group structure. We want to extend any invertible localsystem L0 on J0

C to a local system L on JC , which is compatible with the multiplicationm on JC . However, this actually requires that any invertible local system on J 0

C becompatible with the multiplication on J0

C . Fortunately, this is true in the following sense.For the multiplication m0 : J0

C×J0C → J0

C map one has a natural isomorphism

(m0)∗L0 ∼= L0L0,

of local systems on J0C×J0

C . In terms of fibers of the local system at points L.M, L⊗M ∈J0

C this means a family of isomorphisms (continuous in L, M)

L0L⊗M

∼= L0L⊗L

0M .

For a proof recall that J0C is a quotient of a vector space by a lattice H0(C, Z) and therefore

the map of fundamental groups induced by m0 is the addition in H1(C, Z). This gives therequired isomorphism of local systems on J0

C×J0C .

Now it is clear what the characterization should be:

• (i) The canonical extension L of an invertible local system on J 0C to JC should

have the property

m∗L ∼= LL.

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• (ii) As C generates JC , this is equivalent to an isomorphism on C×JC

(C×JCm−→ JC)∗L ∼= L|CL.

In (ii) we think of C as a subvariety of J1C via the map

AJ1 : C→J1C , AJ1(a) = O(a), a ∈ C.

So, (ii) is a family of isomorphisms of fibers

LL(a)∼= Ea⊗kLL, L ∈ JC , a ∈ C.

1.3.7. Hecke property. This is just the explicit form of the above characterization(ii). We start with an invertible local system E on C and consider the correspondinginvertible local system L0

E on J0C . The canonical embedding AJ1 : C→J1

C induces

π1(C)ab∼=−→π1(J

1C). Therefore, if we use a choice of some L ∈ J1

C to construct an

isomorphism J0C

∼=−→J1

C , M 7→ M⊗L, we get a noncanonical embedding C→J 0C which

again induces π1(C)ab∼=−→π1(J

0C). This means that the corresponding restriction of L0

E toC is isomorphic to E .

So the “simplified” form (ii) of the compatibility condition says

(C×JCm−→ JC)∗L ∼= EL, i.e., LL(a)

∼= Ea⊗LL, L ∈ JC , a ∈ C.

We will restate this in the following terminology :

(1) Each point a ∈ C defines a Hecke modification map Ha : JC → JC , L 7→L(a), hencea Hecke operator Ta = H∗

a on local systems on JC , which is just the pull-back oflocal systems under Ha.

(2) We require the local system L corresponding to E to be an eigenvector for each ofthe Hecke operators Ha, with eigenvalue Ea :

H∗aL∼= Ea⊗kL, a ∈ C.

(3) More precisely, pointwise Hecke modifications fit into a global Hecke modificationmap H : C×JC → JC , and Hecke operator H∗ that takes local systems on JC tolocal systems on C×JC . We require, L to be an eigenvector for H∗ with eigenvalueE :

H∗L ∼= EkL.

1.3.8. Geometric (Unramified) Class Field Theory of Deligne. This is the above corre-spondence E 7→LE = L. We will now give a proof over any closed field k though ourthinking so far has been transcendental and therefore only made sense for k = C.

The importance of general k is for the relation with Number Theory.(4) A technical in-gredient when we deal with general k is that D-modules do not quite work as well, so it

4Actually Number theory would prefer a setting more general then a field!

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is standard to use a different notion of local systems on an algebraic variety over a closedfield – the l-adic local systems in etale topology.(5)

The algebraic construction is given by properties of Abel-Jacobi maps:

Theorem. [Deligne] Let C be a connected smooth projective curve over a closed field k.For each invertible local system E on C there is a unique invertible local system L on JC

which is a Hecke eigensheaf with eigenvalue E , i.e.,

(JCHa:L7→L(a)−−−−−−→ JC)∗ L ∼= Ea⊗ L, a ∈ C.

More precisely, one requires that the above isomorphisms are continuous in a ∈ C, i.e.,that H∗L ∼= EL.

Proof. In one direction, E is the pull-back of L under the embedding AJ 1 : C→J1C .

In the opposite direction one constructs from an invertible local system E on C, a familyof invertible local systems En on Jn

C by

(1) On each Cn one has invertible local system En = E· · ·E .(2) Its “Sn-symmetric part” is an invertible local system E (n) on C(n) constructed by(6)

E (n) def= [(qn)∗E

n]Sn .

(3) Now, for n >> 0 (say n ≥ 2g−2), map AJn is by Abel-Jacobi theorem a projectivebundle. Since π1(P

N) = 0, this implies that E (n) is a pull-back of a local systemon Jn

C which we call LnE .

(4) The family of local systems LnE constructed above have the compatibility with

multiplication property

(JpC×Jq

Cm−→ Jp+q

C )∗Lp+qE∼= Lp

ELqE

and this implies that this family extends uniquely to a local system LE on all ofJC which is compatible with multiplication.

(5) From construction, for n >> 0 we have

Ln+1L(a)

∼= LnL(a) ⊗k Ea, L ∈ Jn

C , a ∈ C.

This implies that the whole LE has this property.

5One can also use D-modules in positive characteristic. The known version of this idea yields aseemingly correct, but more shallow, Langlands correspondence [Bezrukavnikov-Braverman].

6The crucial point of this construction is that the symmetric part of En is again an invertible localsystem. If E is a local system of rank r > 1 then E (n) is a sheaf which is only generically a local system –off the diagonal divisor. However, this is still the basis of Drinfeld’s conjectural construction of unramifiedautomorphic sheaves for GL2. This approach was developed by Laumon into a conjectural constructionfor all GL(r) and Gaitsgory verified Laumon’s conjecture when k = C.

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From this construction of LE the composition E 7→LE 7→LE |C is identity. To see that theother composition L7→ E = L|C 7→ LE is also identity, one observes that the Hecke propertyof L implies that for points ai ∈ C

LO(a1+···+an)∼= Ea1⊗· · ·⊗Ean

⊗LO = E [n](a1,...,an)⊗LO

∼= E (n)a1+···+an

⊗LO,

and for n >> 0 this is the definition of

(LnE)O(a1+···+an) ⊗k LO.

So, for n >> 0 we get natural isomorphisms L ∼= LE on JnC , and then the Hecke property

for both L and LE extends this isomorphism to all of JC .

1.3.9. Geometric Langlands correspondence for Gm. To see the Deligne theorem as a caseof the Langlands correspondence formalism, let us match the above operators Ta withthe expected parameterization of Hecke operators. As the tensoring operators Wa,V onLSG(C) are parameterized by pairs of a ∈ C, V ∈ Rep(G), the correct Hecke operatorsshould have the same parameterization Ta,V a ∈ C, V ∈ Rep(G). Moreover, one shouldhave

(1) Ta,U⊕V = Ta,U⊕Ta,V and(2) Ta,U⊗V = Ta,UTa,V ,

since this is true for tensoring operators. Observation (1) implies that it is sufficient todescribe these operators for irreducible V ’s.

For G = Gm, Irr(G) = χn; n ∈ Z where χn(z) = zn. Since χp+q = χp⊗χq, byobservation (2) we only need to know Ta,χ1 and this is the above operator Ta the pullback of local systems under the map Ha(L) = L(a), L ∈ JC . The more general Heckeoperators do not give anything new since Ta,χn

= (Ta,χ1)n = (Ta)

n, this is the pull back

under the modification map Hn·a(L)def= L(n·a).

1.4. Hecke modifications and the loop Grassmannian. Let us summarize our ex-pectations for Hecke operators:

• (i) Hecke operators should have parameterization Ta,V with a ∈ C, V ∈ Rep(G).• (ii) They should satisfy

Ta,U⊕V = Ta,U⊕Ta,V and Ta,U⊗V = Ta,UTa,V .

So, the basic operators Ta,V are parameterized by a point a ∈ C and V ∈ Irr(G).• (iii) Ta,V should be given by modifications of G-torsors at a which are of “type

V ”.

This subsection examines the modifications of “type V ” at a point a.

We start with the Hecke stack H of all modifications of a G-torsor at a point of the curve.It is a correspondence over BunG(C), so any substack gives an operator on D-modules on

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BunG(C). More generally, any D-module K on H gives an integral operator with kernelK:

IK(F)def= q?[K ⊗OH

p?−]

where we denote by p, q : HBunG(C) the canonical projections and by p?, q? the operationsof a pull-back or direct image for D-modules. This formalism will produce the Heckeoperators.

The restriction Ha of the Hecke stack to modifications at a fixed point a ∈ C will turn outto be equivalent to a simpler object, the so called loop Grassmannian Ga which appears invarious parts of mathematics. For instance, from the group-theoretic point of view Ga isa partial flag variety of a loop group. One can also think of Ga as the simplest nonabeliancohomology, hence the simplest nonabelian motive. The fact that Hecke operators at eachpoint a ∈ C should form an “algebra” (actually a tensor category), will correspond to acertain (obvious) algebraic structure on H and each Ga.

(7)

Loop Grassmannian is the object that we are really interested in. It the next we will seethat it has a stratification by Irr(G) and this will make sense of “Hecke modifications oftype V ” for V ∈ Irr(G).

1.4.1. Hecke stack. Any line bundle L has a canonical modification L(a) at a point a,however this is not true for vector bundles. The effect will be that the Hecke modificationmaps L7→L(a) on line bundles will be replaced by Hecke correspondences for general G.

We define the space Ha,P of all modifications of a G-torsor P at a point a as the moduli of

all pairs (Q, ι) of a G-torsor Q and an isomorphism ι : P∼=−→Q on C−a, i.e., off the point

a. For instance, for a line bundle L this space Ha,L is a discrete set of all L(n·a), n ∈ Z.

More globally, we define the Hecke moduli H as the moduli of quadruples (P, Q, a, ι) of

two torsors P, Q on C, a point a ∈ C and an identification ι : P∼=−→Q off the point a. The

fiber at a given point a ∈ C is the union of all modification spaces at a

Hadef= ∪P∈BunG(C) Ha,P .

The map H → BunG(C)2 makes the Hecke stack a correspondence over the spaceBunG(C), which for each fixed a ∈ C is a loop Grassmannian bundle over the first copyof BunG(C).

1.4.2. Locality. Now notice that the construction Ha,P is local in C – since we allow onlymodifications at a, the moduli will not change if we replace C by any neighborhood U ofa and P by the restriction to U . Actually, the same is true for the formal neighborhooda of a in C. Since any P trivializes on a, this reduces understanding of Ha,P to the casewhen P is the trivial G-torsor G = C×G.

7Ha is the “action groupoid” of a “group” Ga.

13

1.4.3. Loop Grassmannians. We define the loop Grassmannian (or “affine Grassman-

nian”) for (C, a, G) as the moduli Gadef= Ha,G of all modifications at a of the trivial

torsor G. So, this is the moduli of pairs (Q, ι) where Q is a G-torsor on C and ι is thetrivialization (equivalently, a section) of Q off a.

1.4.4. Cohomological interpretation. First, the moduli of G-torsors BunG(C) on C is thefirst nonabelian cohomology H1(C, G). We can think of it as the prototype of a nonabelianmotive – the case G = Gm gives the prototype JC of abelian motives.

The additional information of a trivialization off a means that we are looking at coho-mology classes supported at a. So, the loop Grassmannian is the first local cohomology ata

Ga = H1a(C, G).

(One can also view it as compactly supported cohomology H1c (a, G) on the local curve

a.) This is therefore a local version of BunG(C), hence the simplest nonabelian motive.It packs a number of structures which can be summarized by

Theorem. Loop Grassmannian is fascinating.

Question. What is the meaning of the second local cohomology of G at a point of asurface?

1.4.5. Principle: “Odd cohomologies are hyperkahler ”. The above expression for Ga usesthe cohomology in algebraic geometry. If we work over the complex numbers then wethink of Ga as H1,0

a (C, G). The total De Rham local cohomology then turns out be thecotangent bundle

H1DR,a(C, G) ∼= T ∗Ga.

This is the local version (near a) of the moduli T ∗BunG(C) of Higgs pairs.

Recall that the odd abelian De Rham cohomology of a Kaehler manifold is naturally ahyperkahler manifold [Simpson]. This principle remains true for the first nonabelian coho-mology. Indeed, the global Higgs bundle moduli T ∗BunG(C) = H1

DR(C, G) is hyperkahler(result of Hitchin), and the local Higgs moduli T ∗G is also hyperkahler (result of Don-aldson).

Moreover, the appearance of T ∗BunG(C) in the Beilinson-Drinfeld and Kapustin-Wittenapproaches to the geometric Langlands correspondence, can now be stated as study ofH1,0 using the larger context of H1

DR.

Question. The moduli of local systems should be certain Deligne cohomology?

1.4.6. Loop Grassmannian as a uniformization of BunG(C). We will see that for a fixedG the moduli BunG(C) for all curves C are group quotients of the loop Grassmannian G

for G (and in many ways). For a point a ∈ C denote Goutdef= Map(C − a, G).

14 MANU CHAO

Lemma. If G is semisimple then

BunG(C) ∼= Gout\G.

Proof. If G is semisimple the canonical map Ga 3 (Q, ι)7→ Q ∈ BunG(C) is surjectivesince each torsor trivializes off a point. Then it is clearly a Gout-torsor since it acts simplytransitively on the sections of a torsor over C − a.

(1) (G-bundles on P1.) Let a = 0 ∈ P1 then Gout is G(C[z−1]), hence

BunG(C) ∼= G(C[z−1])\Ga∼= G(C[z−1])\G(C((z))/G(C[[z]]).

Therefore, the set of isomorphism classes of G-bundles on P1 is the set of G(C[z−1])-orbitsin Ga. We will see that it is indexed the same as the set of G(O)-orbits in Ga, i.e., byW\X∗(T ) ∼= Irr(G).

1.4.7. Group theoretic view of the loop Grassmannian. At a point a ∈ C denote by O andK the functions on the formal neighborhood a and on the punctured formal neighborhooda. In terms of any local parameter z at a these are the formal power series O = k[[z]] andthe formal Laurent series K = k((z)). We call G(K) the loop group and G(O) the discgroup at a.

Lemma. Ga∼= G(K)/G(O).

Proof. Let us enlarge Ga to a moduli Ga of triples (Q, ι, η) where Q is a G-torsor onC and ι and η are respectively trivializations of Q off a, i.e., on C − a and near a,i.e., on a. In other words we consider G-torsors on C glued from trivial torsors onC − a and on a. These are given by transition functions on the intersection a, i.e., by

Ga∼= Map(a, G) = G(K). Since G(O) acts simply transitively on all choices of η, we have

G = Ga/G(O) = G(K)/G(O).

1.4.8. Loop Grassmannian of GLn. We say that a rank n lattice is an O-form of Kn, i.e.,

an O-submodule L⊆Kn such that L⊗OK∼=−→ Kn. Equivalently, L has an O-basis which is

a K-basis of Kn.

Lemma. For G = GLn , loop Grassmannian Ga is the moduli of rank n lattices.

Proof. G(K) = GLn(K) acts transitively on K-bases of Kn, hence also on the set oflattices. However, the stabilizer of the trivial lattice On is clearly GLn(O) = G(O).

We can see this in a more intrinsic way. Ga is the moduli of G-torsors, i.e., rank n vectorbundles L on a with a trivialization on a. Taking the global sections this means free rankn O-modules L with the identification L⊗OK ∼= Kn.

1.5. Hecke operators and the stratifications of the loop Grassmannians.

15

1.5.1. Loop groups G(K). Let O = k[[z]]⊆K = k((z))⊇O− = k[z−1]. Over the field ofconstants k, O can be viewed as a group scheme while O− and K are group-indschemes.Similarly, the group G(O) can be viewed as the group of k-points of a group scheme GO

over k, while G(K) = GK(k) for a group ind-scheme GK over k.

For computational purposes we choose a Cartan subgroup T⊆G and two oppositeBorel subgroups B = TN and B− = TN−. Denote by X∗(T ) = Hom(T, Gm) andX∗(T ) = Hom(Gm, T ) the groups of characters and cocharacters of T . As T ∼= Gm

n andHom(Gm, Gm) ∼= Z we have X∗(T ) ∼= Zn ∼= X∗(T ). However, the natural relation is

X∗(T )∼=−→ HomZ[X∗(T ), Z]. Denote by W the Weyl group NG(T )/T .

Loop group GK (almost) lies in the class of Kac-Moody groups which has structure theoryparallel to the standard structure theory of reductive algebraic groups. More precisely,

GK is close to a Kac-Moody group G. Let R = Gm be the group of rotations of theinfinitesimal disc Spec(O), so R acts on O⊆K by s(z) = s−1·z, s ∈ R, and therefore also

on GK and GO. Denote by Gaff the semidirect product GKnR, then G is a certain central

extension 0 −→Gm −→G −→Gaff −→0. In the end, while the Kac-Moody structure theory

applies literally only to G, groups GK and Gaff are close enough so that the notions ofCartan and Borel subgroups, Weyl groups and partial flag varieties make sense for allthree groups.

The role of a Cartan subgroup in GK is played again by the constant Cartan subgroupT⊆G⊆GK, however a more useful version is a Cartan subgroup Taff = T n R = T×Rin Gaff . A new ingredient for loop groups is that they have three basic kinds of Borelsubgroups: Iwahori subgroup I, negative Iwahori I− and semi-infinite Iwahori J (actually,these three constructions can be combined to produce more types of Borel subgroups).Here

Idef= (GO

z 7→0−−→ G)−1B, I− def

= (GO−z 7→∞−−−→ G)−1B−,

while

Jdef= the connected component of BK = TONK.

1.5.2. Schubert cells in the loop Grassmannian. Loop Grassmannian G is a partial flagvariety of GK and we will consider three kinds of Schubert cells, We start with a mapX∗(T ) → G. First, the cocharacters (i.e., the cocharacters of T ) embedded into the loopgroup X∗(T )→T (K), by restricting cocharacters to the punctured formal neighborhood of∞ in Gm. Therefore, the composition X∗(T )→T (K)⊆G(K)G(K)/G(O) = G attachesto each cocharacter λ a point of the loop Grassmannian that we denote Lλ. Now,

Lemma. (a) X∗(T ) embeds into G,

(b) X∗(T )⊆G is precisely the fixed point set GT = GTaff .

(c) Any λ ∈ X∗(T ) generates orbits

• I·Lλ of finite dimension,

16 MANU CHAO

• I−·Lλ of finite codimension, and• J ·Lλ which is semi-infinite (i.e. of infinite dimension and codimension).

This gives a parameterizations of orbits of each of the groups I, I−, J by X∗(T ).

(d) GO-orbits in G are parameterized by Weyl group orbits in cocharacters by

λ7→Gλdef= G(O)·Lλ = ∪w∈W I·Lwλ.

The analogous claim also holds for GO−-orbits.

Proof. The parametrization in (c) follows from (a) and (b) since in a partial flag varietyorbits of a Borel are indexed by fixed points of the Cartan. Claim (d) follows from (c)since W is represented in G⊆GO.

Corollary. GO-orbits and GO−-orbits are in a canonical bijection with Irr(G).

Proof. The combinatorial characterization of the dual group G says that G should containa Cartan subgroup t such that

• there is an identification X∗(T ) ∼= X∗(t) and• this identification identifies the root system ∆t(G) with the dual of the root system

∆T (G).

In particular the Weyl groups are canonically identified.

Now, irreducible representations of G are parameterized by dominant weights

X∗(t)dom⊆X∗(t). However, X∗(t)dom

∼=−→X∗(t)/W since dominant weight form a section

for the W -action. So,

Irr(G) ↔ X∗(t)/W ↔ X∗(T )/W ↔ GO\G.

1.5.3. Irreducible D-modules associated to GO-orbits. We have managed to find a strati-fication of the loop Grassmannian by Irr(G). However, recall that the notion of Heckeoperators at a given point of the curve is categorical – these “operators” form a tensorcategory which is a copy of Rep(G) (1.2.3 and the beginning of 1.4). So we really needto create on G an abelian category equivalent to Rep(G), so that its irreducible objectscome from GO-orbits.

For this we will have to recall that in the realm of D-modules (or perverse sheaves),it is well known how to associate an irreducible D-module to an irreducible subvariety.This will lead to the following abelian category associated to the stratification of G byGO-orbits:

PGO(G)

def= GO-equivariant holonomic D-modules on G.

17

Here a D-module M on X is holonomic if it is “of the least possible size”, i.e., for any Mthe characteristic variety Ch(M)⊆T ∗X is coisotropic and holonomicity is the requirementthat it be Lagrangian.(8)

For an inclusion of smooth subvarieties Yj

→X, OY is a D-module on Y the direct imagej?OY is then a D-module on X and

Lemma. (a) If Y is connected then j?OY has a unique irreducible submodule LY . Actually,LY →j?OY is equality on the neighborhood X − ∂Y of Y .

(b) LY only depends on the irreducible subvariety Y .

Examples. (1) If Y is closed then LY = j?OY .

(2) If Y is open then j?OY is just the sheaf theoretic direct image j∗OY and LY is thesubmodule OX⊆j?OY .

(3) In general, Y acquires singularities along ∂Y and then LY reflects singularities.

(4) The perverse sheaf of LY is the intersection cohomology complex IC(Y ).

(5) The proof of the lemma is based on the following facts

• (i) holonomic D-modules have finite length,• (ii) holonomicity is preserved under j∗,• (iii) j∗OY has no submodules supported on ∂Y .

Here part (ii) is is nontrivial, it uses the notion of b-functions.

1.5.4. Realization of the category Rep(G) on the loop Grassmannians of G. Abelian cat-egory PGO

(G) of GO-equivariant holonomic D-modules on G contains irreducible objects

I !∗λ

def= LGλ

associated to GO-orbits Gλ in G. The parametrization is really by W -orbitsWλ in X∗(T ) and we think of it as a parameterization by Irr(G). Actually,

Theorem. [Drinfeld et al] PGO(G) is canonically equivalent to Rep(G).

Remark. (0) So far, this formulation is not quite meaningful since we have not yet definedG, we only characterized its isomorphism class (in the proof of theorem 1.5.4). As we willsee, the construction of G from G is a part of the more precise statement of the theorem.

Proof. The idea of the proof is that a category that is equivalent to a category of rep-resentations has to have all structures and properties that a category of representationshas. Clearly, Rep(L) comes with tensoring U⊗V , duality U ∗ and with a forgetful functor

8Recall that Gr(DX ) ∼= OT∗X so one can think of DX as a quantization of T ∗X . Then Ch(M) is aclassical limit of M – M has an OX -filtration compatible with with the natural filtration of DX and suchthat Gr(M) is a coherent sheaf on T ∗X , then Ch(M) is the support of Gr(M), independent of the choiceof a filtration.

18 MANU CHAO

F : Rep(L)→ Vec, and these satisfy a number of properties. The formalism of Tannakianduality asserts that any abelian category (R,⊗,−∗,F) with analogous structures whichsatisfy certain list of properties, is equivalent to the category of representations of a groupAut(F) of automorphisms of the functor F (called the fiber functor).

The fiber functor here is the De Rham functor DRdef= RHomDG

(OG,−) of taking the flatsections. If we worked with perverse sheaves instead, then the fiber functor would be justthe total cohomology H∗(G,−).

The interesting part is the “tensoring” operation ⊗ on PGO(G) which we denote by A∗B.

It is constructed as a convolution operation or as a fusion operation.

1.5.5. Convolution. Let A be a subgroup of a finite group B then the space of A-bi-invariant functions H = CAA(B) = C[A\B/A) acquires the structure of an algebra oftencalled a Hecke algebra. If A = 1 this is the group algebra C[B] with the convolutionoperation

f ∗ gdef= (G×G

m−→ G)∗(f⊗g), i.e., (f ∗ g)(b) =

∑

(x,y)∈B2 , xy=b

f(x)g(y).

For biinvariant functions there are some uninteresting repetitions from the diagonal A-action a(x, y) = (xa−1, ay), when we eliminate these we still get an associative algebrastructure on H with (f ∗g)(b) = 1

|A|

∑(x,y)∈B2, xy=b f(x)g(y). To sheafify this, we present

it in a maximally geometric way by the diagram

A\B/A×A\B/Aπ←−A\B ×A B/A

m−→ A\B/A, f ∗ g = µ∗π

∗(f⊗g);

where ×A denotes the product divided by the above diagonal action of A and m is afactorization of the multiplication m.

Now think of PG)(G) as the category P(GO\GK/GO) of D-modules (or perverse sheaves)on the stack GO\GK/GO. Then the formula above interpreted in the world of D-modulesgives an operation on this category

F?Gdef= µ?π

?(FG).

This straightforward geometrization of Hecke algebra construction provides the “tensor-ing” operation on PGO

(G) needed above. It turns out that in this approach the commuta-tivity of the operation is not obvious this prompted Drinfeld to find a second constructionfor which commutativity is clear.

1.5.6. Fusion. Here A ∗ B is constructed as a certain degeneration of A⊗B/ For this we

define for any curve C ind-scheme GCn over Cn with fibers G(a1 ,...,an)def= H1

a1,...,an(C, G),

i.e., over Cn we have a tautological subscheme Tn of C, i.e., Tn→CCn, then GCn is thefirst relative cohomology group H1

Tn/Cn(C×Cn/Cn, G) with support along Tn.

19

Notice that by our definition the fibers of GC2 are G(a,a) = Ga and G(a,b)∼= Ga×Gb for b 6= a

(multiplicativity of cohomology with respect to the support). None the less,

Lemma. GCn is flat over Cn.

Proof. This of course could not happen for finite dimensional schemes since G2 would belarger then G. However, here we deal with indschemes and then flatness means “inductivesystem of flat subschemes”. This is not difficult to arrange, what happens is that asa1, .., an approach a, the product of closures of orbits G(a1,λ1)×· · ·Gan,λn

⊆ G(a1 ,...,an)⊆GCn

approaches the closure of a single but larger orbit Ga,λ1+···+λn⊆ G(a,...,a)⊆GCn .

Once we choose a formal local parameter za at a on C the loop Grassmannian Ga gets

identified with Gdef= G((z))/G[[z]]. This gives identification PGO

(Ga) with the standardrealization PG[[z]](G). This identification turns out to be independent of the choice ofza because GO-equivariant sheaves on G are automatically equivariant under Aut(a =Aut(O).(9)

This allows us for any A,B ∈ PGO(G) to put A⊗B on GaGb = G(a,b) for a 6= b. Now, in

D-modules there is a limiting construction, the nearby cycle functor Ψ.

1.5.7. Such construction also exists in perverse sheaves and in K-theory (called simplyspecialization map), but not in the realm of coherent sheaves. The effect of this is thatthere is a convolution on Db[CohGO

(P)] but it is not commutative (recall that only thefusion is manifestly commutative), however it is commutative on the level of K-groups(because in K-theory the convolution can be identified with fusion). We will use Ψ toproduce from a “constant” family of D-modules AC2−∆C

B on the part GC2 |C2−∆Cof GC2

that lies above C2 −∆C , a D-module A ∗C B on the restriction pf GC2 to the diagonal :

A ∗C Bdef= Ψ(AB).

This is a “constant” D-module on GC2 |∆C∼= GC , so it corresponds to a D-module A ∗ B

on a single G. In other words,

A ∗ Bdef= lim

b→aAB.

Remarks. (a) One can think of the limiting process as a collision of perverse sheavespositioned at points a and b. So, if we think of GO-equivariant irreducible perverse sheaveson G as “elementary G-particles”, i.e., some kind of particle like objects produced from G,then we can say that we have produced the group G and the category of its representationsfrom the study of collisions of G-particles. This is exactly the logic followed in physicsa wile ago when they made sense of collisions of particles by finding a group U whoseirreducible representations can be made to correspond to particles, so that the procedureof producing new particles through collisions corresponds to tensoring of representationsof U and decomposing the tensor product into irreducible pieces.

9Intuitively this is so because Aut(O) preserves GO-orbits, but that’s not the whole story.

20 MANU CHAO

(b) The property of multiplicativity of cohomology with respect to the support, Ga,b∼=

Ga×Gb for b 6= a says that the local cohomologies we find at a and b are independent.This sounds like locality principle in Quantum Field Theory – measurements at pointsseparated in spacetime (by the light cone), are independent. Actually, our situation isthe simplest case of of application of locality, the holomorphic two dimensional conformalfield theory.

The mathematical approximation of the the holomorphic two dimensional conformal fieldtheory is given by the notion of vertex algebras. Then G or rather all GCn together) forma “geometrized vertex algebra” and one gets standard vertex algebras by taking sectionsof certain D-modules on G.

(c) The above construction fusion construction of tensor category uses group G as sheafof coefficients in cohomology. In order to state it without G observe that Ga = H1

a(C, G)can be viewed as cohomology for a pair H1(S, C − a; G) or simply as H1(S, G) for thespace S defined (in homotopy a la Voevodsky) as the quotient of C which contracts C−ato a point. Space S is independent of C and a. Inside of C2 we consider the union A ofcurves C×a and ∆C and we have a pair (C2.C2−A) which maps to C by pr1. Let C → Cbe the corresponding quotient of C2 by contracting the fibers of C2−A. The central fiberof C → C is S while the general fiber is the cowedge S∨S. So, C is a cobordism of S∨Sand S, so it is something like a (co)group structure on S in cobordism category. This isthe structure which produced fusion once we put in the coefficient group G.

Proposition. The fusion construction ∗ coincides with the convolution construction ∗.

Proof. As a bridge between the two constructions one extends the convolution constructionto the C2-setting that houses the fusion construction. Then the main observation is thatthe fibers in the convolution map grow slowly (it is a “semismall map”).

1.5.8. Hecke operators. Now we can complete the picture. At each point a ∈ C thecategory PGO

(G) is a copy of the category Rep(G), and we will see that PGO(G) acts on

the category P(BunG(C)) of D-modules on BunG(C) by convolution/fusion. These arethe Hecke operators.

The simplest point of view is group theoretical, we can think of

P(BunG(C)) = P(Gout\Ga) = PGout(Ga)

as the category of Gout-equivariant D-modules on Ga. However, the above constructions ofA∗B do not require that A be GO-equivariant (but then the result will not be equivarianteither). So either of these constructions yields an action of the tensor category PGO

(G)on the category P(G) of all D-modules on G, and this preserves the natural subcategoryPGout

(G)⊆ P(G).

21

In terms of the fiber functor F : PGO(G) → Vec, F(M) = RHomDG

(OG, M), the Heckeproperty of F ∈ P(BunG(C)) says that for any A ∈ PGO

(G) one has

F ∗ A ∼= F(A)⊗k F .

It suffices to check this for a collection ofA’s that generate the tensor category PG(GO), forinstance the irreducibles LGλ

. For λ ∈ X∗(t) denote by L(λ) the irreducible representationof G with extremal weight λ, then the Hecke property of F ∈ P(BunG(C)) reduces to

F ∗ LGλ∼= L(λ)⊗k F , λ ∈ X∗(T )/W.

One can also us just the irreducibles corresponding to the fundamental weights etc.

1.6. Examples in type A. For G = GLn, take T to be the diagonal Cartan so canon-ically T ∼= (Gm)n and X∗(T ) ∼= Zn. In terms of the standard basis ei of Kn, the latticeLλ corresponding to the cocharacter λ ∈ Zn is then ⊕n

1 z−λiO·ei. The GO-orbit Gλ thenconsists of all lattices L which are in the same position to the trivial lattice On as thelattice Lλ in the sense that the dimensions dim[(L + zp·On)/zpOn], p ∈ Z, are the sameas for Lλ.

In particular, for ωi = (1, ..., 1.0, ..., 0) ∈ Zn − X∗(T ) with i ones, 1 ≤ i ≤ n, we getG(O)-orbits

Gωi

def= lattices L; z·On⊆L⊆On and dim(L/z·On) = i ∼= Gri(n).

Elementary Hecke modifications for GLn.

1.7. Langlands conjectures in number theory.

1.7.1. Class Field Theory. One could say that the the structure number theory numbertheory attempts to describe is that of the totality of all of finite extensions of Q, i.e.,the fields of algebraic numbers. Equivalently this can be viewed as the description of theGalois group GalQ of Q, or of the category of finite dimensional representations of GalQ.

The Class Field Theory resolves the abelian part of the problem, i.e., it describesthe largest abelian quotient GalQ

ab or equivalently its irreducible (i.e., one dimensional)representations.

1.7.2. Langland’s conjectural nonabelian generalization of Class Field Theory. Langlandsconjectures are an attempt to deal with the nonabelian nature of GalQ. From the newpoint of view, the n-dimensional representations ρ of GalQ, i.e., group maps GalQ →GLn(C), correspond to irreducible automorphic representations πρ of GLn(AQ).(10) TheClass Field Theory is indeed the case n = 1 of this prediction.

10It is actually better to allow ρ to be in a larger class of representations of the so called Weil group

WQ which is an amplified version of GalQ and has GalQ as a quotient.

22 MANU CHAO

Here AQ, the ring of adels, is a large topological ring which is associated to Q. It containsQ and one can think of it as a “locally compact envelope of Q”. This locally compactsetting allows us to use analysis based on the Haar measures on locally compact groups.

Qualification “automorphic” means that the representation πρ appears as a partof the large representation of GLn(AQ) on functions on its homogeneous spaceGLn(Q)\GLn(AQ).

1.7.3. L-functions. One important feature of the correspondence ρ7→πρ should be that itpreserves the basic numerical invariants – the L-functions

Lπρ(s) = Lρ(s), s ∈ C.

L-functions are maybe the closest among mathematical objects to the Feynman integralsin physics. The behavior of automorphic L-functions is understood much better. So, if wewould know that the L-functions of Galois representations are also L-functions of someautomorphic representations we would obtain deep information on behavior of L-functionsof Galois representations. For instance, in this way Fermat conjecture has been reducedto a special case of Langlands conjectures.

1.7.4. Unramified automorphic representations. A particularly nice class of automorphicrepresentations are the unramified representations. A nice feature of such representationsis that they contain a distinguished vector, i.e., a distinguished function which is aneigenfunction of certain Hecke operators. Functions of this kind appear in the classicalcomplex analysis as automorphic functions. From the adelic point of view automorphicfunctions are the functions on GLn(Q)\GLn(AQ)/GLn(OQ) where GLn(OQ) is a canonicalmaximal compact subgroup of GLn(AQ). So, the unramified automorphic representationsare classified by automorphic functions.

There is also the notion of unramified representations of GalQ . These representations aredetermined by what they do on particular elements of GalQ which act as Frobenius auto-morphisms at points of Spec(Z). The unramified part of the global Langlands conjecturethen says that to each unramified n-dimensional representation ρ of GalQ there corre-sponds an automorphic function fρ with the eigenvalues of Hecke operators prescribed bythe values of ρ on Frobenius automorphisms.

1.7.5. Langlands duality of reductive groups. However, automorphic functions have clas-sically been studied for various reductive groups, not only for GLn. Incorporating thisLanglands predicted that for any reductive complex group G, group maps GQ → G(C)correspond to irreducible automorphic representations of G(AQ) where G is another re-ductive group called the Langlands dual of G. The case of GLn is recovered since forG = GLn the dual group G is again isomorphic to GLn.

The duality operation G7→G is simple on the combinatorial level (i.e., if we only wantto produce an isomorphism class of groups rather then a specific group). Then it is acombination of duality for root systems and duality for lattices (groups isomorphic to

23

some Zn). However, a functorial construction G7→G is deeper and more recent, the onlyknown construction (Drinfeld, Ginzburg,...) uses loop groups.

1.7.6. Attraction. If we restrict ourselves to G = GLn, we may expect that the conjectureshave some reasonable explanations. However, the general case suggests that the theoryis much deeper since the relation between G and G is deep. At this point we see thatwe are confronted with a mystery of the universe which chooses to present the samefacts sometimes in the G form and sometimes in the G form, though to us the two seemunrelated. It is as if you find yourself in a boat in a sea during a lively storm, andstimulated by the danger to your existence and a wild beauty of the elements, you arepushed into a strong, if irrational, conviction of the unity of the sea and the wind whichto the uninitiated may seem to only meet accidentally. And so you ask yourself for aFourier transform that exchanges the wind and the sea.

The G–G phenomena turns out to be of general importance. This has indeed been con-firmed by appearance of this duality in physics as Montonen-Olive duality generalizingthe classical electro-magnetic duality, and in various other string related phenomena suchas Seiberg-Witten.

1.7.7. Curves over finite fields. The Class Field Theory and the Langlands program gobeyond Q itself and deal with a class of fields F called “global fields of dimension one”.These fall into the

(1) arithmetic case: all algebraic number fields F ,(2) geometric case: transcendence degree one extensions F/F of finite fields F.

The first case is much deeper, it includes F = Q and provides a larger setting by studyingall finite extensions of Q at the same time and with the same ideas.

The second case is much simpler since such field is a field of fractions F(C) of a uniquesmooth projective algebraic curve C over this finite field F. So, number theoretic objectshave geometric interpretations and one can use the methods of algebraic geometry.

From the point of view of the more traditional arithmetic case, the geometric case isa “baby case” – the progress is faster and we gain intuition about the arithmetic case.However, the geometric case is also important as a bridge from number theory to algebraicgeometry and other mathematics. It depicts number theory as the deepest study ofone dimensional geometry with lessons to be applied to algebraic geometry, differentialequations, and more recently to physics.

In the remainder we will now review quickly how the (so called) Geometric LanglandsConjectures arises from the Langlands program in Number Theory applied to the geomet-ric case where Q is replaced by the field F(X) of rational functions on a curve X definedover a finite field.

24 MANU CHAO

1.7.8. “Functions-sheaves” dictionary of Grothendieck. This is the mechanism underlyingDrinfeld’s “geometric” approach to Langlands conjectures, which produces the conjecture1.1.3(b). (Also a basis of much else.) The idea is that in a number of settings certainclass of sheaves S behaves as a categorification of a certain class S of functions. The twoclasses happen to be closely related but while functions form a vector space S, the sheavesare one categorical step further since they form a category S.

The basic example is the situation considered by Grothendieck. The geometric object is analgebraic variety X defined over a finite field F with q elements. One considers the derivedcategory Db

constr(X) of constructible l-adic sheaves in etale topology on X = XF thecorresponding variety over the closure F of F. The Galois group GalF acts on X and the

fixed points XGalF

are precisely the F-points X(F) of X. A sheaf F ∈ Dbconstr(X) is said

to be defined over F if it is equivariant under the action of the Galois group GalF. SinceGalF is generated by the Frobenius automorphism FrF, equivariance structure means just

an isomorphism φ : (FrF)∗F

∼=−→F of the Frobenius twisted sheaf and the original sheaf.

Now, any sheaf F defined over F defines a function χF on the set X(F), at a Frobeniusfixed point a ∈ X(F ), φ acts on the stalk Fa since (Fr∗FF)a = FFrF(a) = Fa and

χF (a)def= Tr[φ : Fa → Fa].

Moreover, One actually gets a sequence of functions where χnF is defined on X(Fn) the

points of X over the degree n extension Fn of F, using the trace of φn.

The main thing about this formalism is that it is functorial under maps f : X → Y , i.e.,

χf∗G = (fF)∗χGandχf!G = (fF)!χF ,

here fF denotes the restriction f : X(F)→ Y (F), f ∗ and (fF)∗ denote the standard pull-

backs of sheaves and functions, while f! is the proper direct image of sheaves and (fF)!

denotes the direct image of functions by taking sums over fibers.

How much does the system of functions record about F? It determines the class of F inthe Grothendieck group of Db

constr(X) (Chebotarev-Laumon).

While the passage from sheaves on XF to a system of functions on all X(Fn) is straight-forward, the reverse procedure of geometrization is an art. One starts with an interestingfunction f on X(F) and looks for a sheaf F such that χF = f . The main idea is thatif f is built from simpler functions using the functoriality of functions the correspondingfunctoriality of sheaves should be used to build F .

1.7.9. Geometrization of the (geometric case of) unramified global Langlands conjectures.Recall that the unramified Langlands conjecture predicts that to each unramified rep-resentation ρ of the Galois group GF of a global field F of dimension one there shouldcorrespond an automorphic function fρ on G(F )\G(AF )/G(OF ), i.e., an eigenvector fornumber-theoretic Hecke operators, with eigenvalues prescribed by ρ.

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When F is the field of rational functions on a curve X defined over a finite field, Galoisgroup G(F ) is roughly the fundamental group of the curve X, more precisely it is a limitof fundamental groups of X−S for finite sets S. So, a representation ρ : GalF G defines alocal system on X−S for some finite S, i.e., a “local system with singularities” on X. Theunramified representations ρ are the ones for which the local system has no singularities.

On the other hand the space G(F )\G(AF )/G(OF ) turns out to be the set BunG(C)(F )of F -points of the moduli of F -bundles on C. Now it is clear what should be the “ge-ometrization”. Instead of constructing a function fρ on BunG(C)(F ) we should constructa sheaf ρ on the stack BunG(C) such that χρ = fρ. This specific conjecture 1.1.3(b) waseventually made more transparent by a more abstract conjecture 1.1.3(a).

1.7.10. Geometric Langlands conjecture vs the original Langlands conjectures. The initialhope was approximately that: (i) sheaves are a much more subtle setting so one will haveto define the problem better, (ii) sheaves are a richer setting with constructions that donot have obvious meaning for functions.

This has not quite worked out. In positive characteristic, i.e., in the geometric case, theproof for GLn completed by Lafforgues did not use this geometrization of Langlands con-jectures (but some other ones which is not called “geometric Langlands”, due to Drinfeldagain).

Another obvious weakness of the geometric conjectures is that they do not even try toaddress the arithmetic case of the original conjectures. I believe that this will be correctedin near future through a (non existing) formalism of “stochastic algebraic geometry”.

However, even in the geometric case the geometric Langlands conjectures reach furtherthen the standard ones. One example is that they are geometric enough to make sensefor complex curves, and this lead to relations with representation theory of loop groupsand vertex algebras, and to the conjectural S-duality in Quantum Field Theory.

1.8. Appendix. Dual group G comes with extra features. This subsection is anamplification of the above results on the loop Grassmannian which has so far not figuredin the Witten approach.

The above construction G7→G is certainly not the only possible. However, this methodproduces G with some extra structures.

Ginzburg G has a canonical choice of a Cartan and Borel subgroup T⊆B⊆G. On the level ofrepresentations, the representations of GZ that we produce from sheaves in PGO

(G)by means of the fiber functor come with a grading and an action of H∗(G).

(1) Moreover, they come with (two) canonical basis.(2) From a complex group G one obtains not only a complex group G but actually the

split form GZ of G over integers. Moreover, for any tensor category T with proper-ties sufficiently alike the properties of the category of modules over a commutative

26 MANU CHAO

ring, the same procedure should give a meaning to G over T . As proposed byDrinfeld and Lurie, this in particular applies to the category of spectra.

For simplicity in this subsection we often refer to perverse sheaves rather then the D-modules because the fiber functor in this case is just the total cohomology. In particularit associates to the irreducible perverse sheaf LGλ

corresponding to the orbit Gλ (this is theso called intersection cohomology sheaf IC(Gλ) of the closure of the orbit), it associatesthe total intersection cohomology of Gλ.

1.8.1. GLn. The basic features of the construction are easily seen in the case of G = GLn.The dual group is again GLn but the geometric construction says more precisely that ifour group G is GL(V ) for a complex vector space V , then the dual group G is naturallyGL(V ) for the complex vector space V = H∗[P(V ), C] which we call the Langlands dualof the vector space V .(11)

Clearly, the topological (actually motivic) realization of V endows it with a Z-form VZ =H∗[P(V ), Z], so the dual group G = GL(V ) comes equipped with a split Z-form GZ =GL(VZ). Moreover G has a canonical Cartan subgroup T (the stabilizer of the grading oncohomology), a canonical Borel subgroup B (the stabilizer of the filtration given by thegrading).

It will turn out that all irreducible representations of G, which we realize as intersec-tion cohomology of algebraic varieties, have canonical bases given by irreducible com-plex subvarieties. For instance the fundamental representations of GL(V ) are L(i) =H∗(Grp(V ), C), i = 1, ..., dimV , and the basis are given by the Schubert varieties inGrassmannians.

1.8.2. Subgroups T⊆B⊆G. For a sheaf M ∈ PGO(G), group G acts on the corresponding

vector space obtained from the fiber functor

F(M) = RHomDG(OG, M) ∼= Ext•DG

(OG, M).

This gives a Z-grading on F(M) and also a compatible action of ExtDG(OG,OG)

∼=−→H∗(G, C).

In particular, we have an action of the hyperplane section class in H2 which is clearlynilpotent. It turns out that the hyperplane section acts a regular nilpotent e in the Lie al-gebra g. Moreover, the action of H∗(G, C) can then be explained through an isomorphismof H∗(G, C) with the enveloping algebra of the centralizer Zg(e)⊆g.

The grading can be viewed as an action of the multiplicative group Gm which turns outto come from an embedding ρ : GM → G. On the Lie algebra level this embedding gives a

11G acts on the fundamental representation which is here realized as the intersection cohomology ofthe closure of the orbit Gω1

corresponding to the first fundamental weight. However, Gω1

∼= P(V ) andsince the orbit is compact the intersection cohomology reduces to ordinary cohomology H∗[P(V ), C]. Thisaction of G on V identifies it with GL(V ).

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semisimple element h ∈ g. The compatibility of the action of H∗(G, C) with the gradingnow says that e, h extends to an sl2-triple e, h, f .

Since a regular nilpotent lies in a unique Borel subalgebra 4, G comes with a distinguishedBorel subgroup B. The semisimple part h of an sl2-triple with a regular nilpotent is itselfa regular semisimple element, so it lies in a unique Cartan subalgebra t. Now, [h, e] = 2eimplies that the corresponding subgroup T lies in B.

1.8.3. Canonical bases of representations. Recall that the closures of GO-orbits Gλ arespecial finite dimensional Schubert varieties in G, however to each ν ∈ X∗(T ) we havealso associated a semi-infinite Schubert cell J ·Lν .

Lemma. (a) For M ∈ PGO(G), there is a canonical decomposition

F(M)def= H∗(G, M) ∼= ⊕ν∈X∗(T ) H∗

c (J ·Lν , M) ∼= ⊕ν∈X∗(T ) H∗J ·Lν

(G, M).

Actually, the summand H∗c (J ·Lν , M) is the ν-weight space of the Cartan subgroup t in

the representation F(M) of G.

(b) In the case when M is an irreducible object, i.e., the intersection cohomology sheafIC(Gλ) of some GO-orbit, the irreducible components Irr(Gλ ∩ J ·Lν) of the intersectionof Gλ with the corresponding semi-infinite Schubert cell J ·Lν , give a basis of both theν-weight space of the corresponding irreducible representation,

Remarks. (0) The existence of canonical ones were introduced by Lusztig. bases of irre-ducible representations is a deep discovery of Lusztig, He introduced two pairs of “canoni-cal” and “semicanonical” bases, the bases above should be related to Lusztig’s semicanon-ical bases.

(1) Part (a) is often called the geometric Satake isomorphism, because it is a categorifi-cation of Satake’s result on p-adic groups.

(2) This is essentially the only known case where genuine intersection cohomology has abasis of algebraic cycles.

(c) Taking the moment map images of the cycles in Irr(Gλ ∩ J ·Lν) Anderson formulateda charming combinatorics of irreducible representations in terms of certain polytopes.

1.8.4. Arithmetic content. The above construction via Loop Grassmannian actually givesa topological interpretation of representations of a split reductive algebraic group Ak overan arbitrary Noetherian commutative ring k of finite global dimension, say Z or Fp. Inthis approach, representations of a reductive group Ak are constructed using the geometryassociated to its Langlands dual group G over complex numbers, so we start with G andwork towards Ak = Gk. In this way, category of Gk-modules has been “localized”, i.e.realized as a category of sheaves. This gives a new setting for the study of modular

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representations of reductive groups (or even the representations over rings), which usescomplex geometry.

We have no way to introduce exotic coefficients in the realm of D-modules, so we nowswitch technologies and on the loop Grassmannian G of a complex reductive group Gwe will consider the abelian category PGO

(G, k) of perverse sheaves with coefficients inmodules over a commutative ring k.

Now notice that over a general coefficient ring there are some subtleties which are sweptunder a rug for k = C. A character of a Cartan subgroup T of the dual group Gk, thenλ defines three Gk-modules with extremal weight λ

W (λ, k)→ L(λ, k)→ S(λ, k).

Here the Schurr module S(λ, k) is constructed geometrically as the global sections of aline bundle on the flag variety of Gk. The Weyl module W (λ, k) is traditionally introducein algebra as a quotient o a Verma module. They are related by S(λ, k) == W (−λ, k)∗.Finally, L(λ, k) is the image of the canonical map W (λ, k)→S(λ, k). When k is a fieldthis is the irreducible representation with extremal weight λ. If k also has characteristiczero then the three constructions coincide.

Theorem. Let k be a commutative ring, noetherian and of finite global dimension.

(a) The abelian category PGO(G, k) together with the convolution/fusion operation ∗ and

a certain duality operation is a Tannakian category with a fiber functor given by theglobal cohomology F = H∗(G,−) : PGO

(G, k) −→k−mod. So it is canonically equivalentto the category of representations of the group scheme Aut(F).

(b) Group scheme Aut(F) is the split form Gk of the dual group G over k.

(c) For any cocharacter λ of T , the corresponding GO-orbit Gλ

j→G defines three perverse

sheaves(12)

I!(λ, k)def=jp

! (kGλ[dim Gλ]) → I!∗(λ, k)

def=j!∗(kGλ

[dimGλ]) → I∗(λ, k)def=jp

∗(kGλ[dimGλ]).

(1)

The associated representations of Gk, i.e., the cohomologies of these perverse sheavesare respectively

W (λ, k)→ L(λ, k)→ S(λ, k).

In particular, L(λ, k) is realized as the standard (“middle perversity”) intersection ho-mology of Gλ with k-coefficients.

Theorem. When k is the ring of integers situation simplifies since I!(λ, Z) = I!∗(λ, Z) andits dual is I∗(λ, Z).

12Once we are not working over Q there are several versions of intersection cohomology sheaves corre-sponding to various perversities.

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1.8.5. Example. In the remainder of this section we consider the simplest example of theconvolution and of the coincidence of intersection homology and algebra over a generalcommutative ring k.

Let G = PGL(V ) for V = C2, the dual group G = SL(V ) has a fundamental repre-sentation L(1) = V = H∗(P(V ), C) and L(1)⊗L(1) is a sum of the trivial and adjointrepresentation L(0)⊕ L(2) = C⊕ sl2(C).

Geometrically, two copies of the representations L(1) are realized as constant sheaves(with a shift [1]) on two copies of G1 = G1 = P(V ) = P1. These P1’s combine in a non-

symmetric way into a P1-bundle X over P1 Along the “zero” section X is the tangentbundle to P1, and along the “infinite” section it is the cotangent bundle.

Next, X = G2 is obtained by blowing down the “infinite” section (a (−2)-line), to getX π−−−→ X. Then for L(n) denoting I!∗(nρ, C),

L(1) ∗ L(1) = CP1[1] ∗ CP1 [1] = π∗CX [2] = CX [2]⊕ Cpt = L(2)⊕ L(0).

The map Xπ

−−−→ X = G2 is a compactification of the Springer resolution T ∗(P1)→ Nof the nilpotent cone N⊆ sl2(C), by adding a line P1 at infinity. The only singularity ofX is at 0 ∈ N and the link of 0 in N is RP3 = S3/Z2. This Z2 is felt in the intersectionhomology over a general coefficient ring k. Denote by k2 (the 2-torsion in k) and k/2k (the2-cotorsion), also the corresponding sheaves supported at the point G0 = 0⊆ N⊆ G2).Then I!(2, k) = kX [2] and it is an extension 0→k2→I!(2, k)→I!∗(2, k)→0,, while I∗(2, k)is also an extension 0→I!∗(2, k)→I∗(2, k)→k/2k→0.

Since X is paved by affine spaces H∗[G, I!(2, k)] ∼= k3, this is the Weyl module W (2, k) =sl2(k) = ( a b

c −a ) , a, b, c ∈ k. Similarly, H∗[G, I∗(2, k)] is its dual S(2, k) ∼= W (2, k)∗ ∼=k3. The canonical map ι : W (2, k)→S(2, k) is the simplest invariant form on sl2(k):

〈x, y〉def= tr(xy). Since

〈( a bc −a ) ,

(a′ b′

c′ −a′

)〉 = 2aa′ + bc′ + b′c,

one has ι ( a bc −a ) = 2aα + bβ + cγ, for a certain basis α, β, γ of S(2, k). So the kernel and

the cokernel of ι are k2 and k/2k.

The fusion version of convolution in our example means the degeneration of the smoothcubic P1×P1 = G1×G2 in P3 to a singular cubic G2.

1.8.6. Remarks. This motivic point of view also comes with explicit geometric construc-tions of the group algebra O(G) and of the positive part U n of the enveloping algebra U g

of the dual Lie g.

E-mail address : mirkovic@math.umass.edu